Question

9
find the length of ef in the trapezoid shown
below.
ef=

Explanation:

Step1: Identify the midline property

In a trapezoid, the length of the midline (EF) is the average of the lengths of the two bases (BC and AD). So, the formula is $EF=\frac{BC + AD}{2}$. Here, $BC = 8x$, $AD = 16$, and $EF = 7x - 1$. Substituting into the formula gives $7x - 1=\frac{8x + 16}{2}$.

Step2: Solve for x

Simplify the right - hand side: $\frac{8x + 16}{2}=4x + 8$. So the equation becomes $7x-1 = 4x + 8$. Subtract $4x$ from both sides: $7x-4x-1=4x - 4x+8$, which simplifies to $3x-1 = 8$. Add 1 to both sides: $3x-1 + 1=8 + 1$, so $3x=9$. Divide both sides by 3: $x = 3$.

Step3: Find the length of EF

Substitute $x = 3$ into the expression for EF, which is $7x-1$. So, $7\times3-1=21 - 1=20$? Wait, no, wait. Wait, maybe I made a mistake. Wait, let's re - check. Wait, the two bases are BC and AD. Wait, BC is $8x$, AD is 16. The midline EF is the average. But also, from the markings, E is the midpoint of AB and F is the midpoint of CD, so by the midline theorem for trapezoids, $EF=\frac{BC + AD}{2}$. So $7x-1=\frac{8x + 16}{2}$. Let's solve this equation again. $\frac{8x + 16}{2}=4x + 8$. So $7x-1=4x + 8$. Subtract $4x$: $3x-1 = 8$. Add 1: $3x=9$, $x = 3$. Then EF is $7x-1=7\times3-1 = 20$? But wait, BC is $8x=24$, AD is 16. The average of 24 and 16 is $\frac{24 + 16}{2}=\frac{40}{2}=20$. Wait, but maybe I misidentified the bases. Wait, maybe BC is the top base and AD is the bottom base. Wait, but let's check again. Wait, maybe the formula is $EF=\frac{BC + AD}{2}$, so if $x = 3$, BC is 24, AD is 16, average is 20. But wait, maybe I made a mistake in the formula. Wait, no, the midline of a trapezoid is the average of the two parallel sides (bases). So if E and F are midpoints of the non - parallel sides, then $EF=\frac{AB_1+AB_2}{2}$, where $AB_1$ and $AB_2$ are the two bases. So in this case, the two bases are BC (length $8x$) and AD (length 16), and the midline is EF (length $7x - 1$). So the calculation seems correct. Wait, but let's check with $x = 3$. BC is 24, AD is 16, midline is 20. So EF is 20? Wait, maybe I made a mistake in the first step. Wait, maybe the two bases are AD and BC, with AD = 16 and BC = 8x, and EF is the midline. So $EF=\frac{AD + BC}{2}$, so $7x-1=\frac{16 + 8x}{2}$. Then $7x-1 = 8 + 4x$. Then $7x-4x=8 + 1$, $3x=9$, $x = 3$. Then EF is $7\times3-1 = 20$. Yes, that's correct.

Answer:

34